Millionaire yoga guru Bikram Choudhury has been accused of rape, sexual assault and "human trafficking" in two new lawsuits filed by former students.

According to the suits, filed in Los Angeles this month, Choudhury has "a propensity to sexually assault young women". The two suits, filed by women named only as Jane Doe 1 and Jane Doe 2, charge Choudhury and Bikram Yoga College of India of sexual battery, false imprisonment, discrimination, harassment and other counts.

Calcutta-born Choudury, the 67-year-old founder of the "hot yoga" chain, runs an international empire of about 650 yoga studios across three continents. Choudhury claims to have taught yoga to presidents Nixon, Reagan and Clinton. Celebrities including Lady Gaga, David Beckham and Madonna have practiced Bikram, a series of 26 yoga poses that take place in rooms heated to over 100F (37.8C).

Another former Bikram student, Sarah Baughn, filed a suit in March claiming Choudhury made sexual advances while she was training to be a teacher. That complaint alleged sexual harassment but not rape. The two new plaintiffs have opted to maintain their anonymity.

All three women are being represented by Mary Shea Hagebols of Shea Law Offices in Oakland, California. "I have worked with the victims of sexual assault for decades," she said. "It is very difficult for someone to come forward and speak out against someone powerful and wealthy." She said her clients wished to protect other women from similar assaults.

Petra Starke, president of Bikram Yoga College of India, said she could not comment on the cases.

Jane Doe 1, who filed her suit on 7 May, claims Choudhury raped her twice in the fall of 2011. She claims that Choudhury's inner circle (identified in the suit as defendants 1-25) were aware of their boss's behaviour.

"Plaintiff is informed and believes that after defendant Bikram Choudhury lures the young women to his place of business, through various schemes and designs and with the participation of his inner circle who are knowledgeable of and complicit in the abuse, he sexually assaults and/or rapes them," the suit alleges.

In March this year Jane Doe 1 said she confronted Choudhury and told him his conduct was "unlawful, unwanted, and unconsented to". She said Choudhury tried to "inflict guilt" on her by saying if she came forward, the Bikram community would suffer.

Jane Doe 2, who filed on 6 May, claims that Choudhury raped her in November 2010 after her boyfriend gave her a gift of a $13,000 nine-week teacher training programme at Choudhury's Bikram Yoga studio in San Diego. The grueling training programme began at 7am and ended at between 2am and 3am. During the sessions, would-be teachers were told what they could eat, wear (skimpy clothing and nothing green), what they could say and "the expressions on their faces", Jane Doe 2 alleges.

During weeks of gruelling training Choudhury singled out Jane Doe 2 claiming she was "a piece of gold in a roomful of brass". His approaches became more sexual and were rebuffed. The suit alleges the exhausted student was raped when she was "too weak and overwrought to fight him off".

The suits paint a cult-like atmosphere at the training camps. Trainees were allegedly told that Choudhury is on the same level as Jesus Christ or Buddha, that Bikram yoga can cure cancer and that practitioners will be able to live to 100 years old. Trainees were bullied and humiliated as well as praised, the suits say, and Choudhury allegedly gave lectures in which he disparaged gays, Americans and made ethnic slurs.

Jane Doe 2 claims that Choudhury recruits volunteers from overseas who are "so in fear of defendant Bikram Choudhury's wrath that they will travel to the US and risk violating immigration laws in order to serve him. Once in the US these volunteers work for little or zero pay. "Their duties include grooming him, massaging him, making his tea, bring[ing] him food and being forced to submit to sexual assaults and rapes against their will," the suit alleges.